homework2

# homework2 - Problem set 2 Fourier transforms δ-functions...

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Unformatted text preview: Problem set 2 - Fourier transforms δ-functions and quantum dynamics due on Friday, September 23rd, in class. (Dated: September 16, 2011) 1. Width of a Gaussian wave packet . In class we talked about the time evolution of a gaussian wave packet. We saw that one can easily define a wavefunction Ψ( x,t ) that has a non-zero momentum expectation value p = ¯ hk in momentum space through: φ k = Ne- ( k- k ) 2 4Δ k 2 (1) (*) Consult ’Lecture5’ notes to answer the following: (a) Find the normalization N of this function. (b) What is the time dependence of φ k ? (c) What is the position representation of the above wavefunction? (d) Find the variance of x , Δ x ( t ) 2 as a function of time, and in terms of the other parameters (Δ k, m, Δ x ). Helpful formula and hints: Z dk e- b ( k- a ) 2 = p π/b (2) as long as Re b > 0 for any a and Im b . To take advantage of this formula you might need to complete the square in some exponent. Note that ψ ( x,t ) will emerge as a complex Gaussian, ψ ( x,t ) ∝ e- α ( x- v g t ) 2 with α complex. | ψ ( x,t ) | 2 will be a real Gaussian. Identify the variance of x from the probability density by inspection . To do this you do not need to integrate over space, just calculate the absolute value square.this you do not need to integrate over space, just calculate the absolute value square....
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homework2 - Problem set 2 Fourier transforms δ-functions...

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